Hardy and Wright, Chapter 4

Following last week’s discussion on chapter 3 of Hardy and Wright’s number theory book, we talked about chapter 4 (Irrational Numbers) this week.

Chris claimed (perhaps quoting something) that Pythagoras is like Shakespeare (or was it Homer?): things with his name as author may not have been authored by him. That’s probably not his exactly quote, but it’s the spirit of it.

We agreed that it was fun to see that roots of monic polynomials with integer coefficients are either integer or irrational. The historical notes about why square roots were only shown irrational (or integral) up to (possibly including) 17 were also interesting.

One of the lemmas in the book is that \log_{10}2 is irrational, and the book states that “\log_n m is irrational if n and m are integers, one of which has a prime factor which the other lacks.” When I came across this note, I wondered if there was some cute way to say this condition, that “one of which has a prime factor which the other lacks.” I thought maybe something with gcds. We talked about it a little, and didn’t come up with much. We discussed whether it meant prime power factor, or just prime factor. So, for instance, are n=2^2\cdot 3 and m=2\cdot 3 handled by the test? Andy pointed out that “having a prime power factor which the other lacks” is simply saying that the numbers aren’t equal, which clearly is too strong a statement here. However, a more general statement, allowing differing prime powers, seems hard (or at least harder) to state. So we left it.

The main thing I was curious about from this chapter was from the end of the chapter, section 4.7, “Some more irrational numbers”, where it is shown that e^{\gamma} is irrational for any rational \gamma and that \pi^2 is irrational. Both of these proofs rely on a function

f(x)=\dfrac{x^n(1-x)^n}{n!}

whose appearance here I would really love to understand. Is it just something terribly clever somebody (I’m looking at you Euler, even if it doesn’t make sense historically (which I have no idea about)) wrote down? Or should I have expected to see it, or even come up with it myself?

In the proof that e^{\gamma} is irrational, they reduce to the case \gamma=h an integer, and then define

F(x)=h^{2n}f(x)-h^{2n-1}f'(x)+\cdots-hf^{(2n-1)}(x)+f^{(2n)}(x).

While sitting down to write this post, I noticed that this comes from trying to integrate h^{2n+1}e^{hx}f(x), so I guess that’s ok. But I still don’t see where the f(x) comes from. Is it just picked out because it has the property that f(1-x)=f(x), and is bounded above by 1/n!? Or perhaps it comes about because you can think of it as the n-th term in the Taylor series for e^{x(1-x)}, as Chris suggested? Why would I care about that Taylor series though? And what does the integral they compute, \int_0^1 h^{2n+1}e^{hx}f(x)\,dx have to do with any of this? Chris suggested perhaps some error term in a Taylor series? Perhaps the Taylor series for e^h (whose irrationality we are trying to show, after all)?

We had a couple of other thoughts (whose they were, I don’t recall). Was this integral some sort of useful convolution? Or perhaps related to the Gamma function? Or maybe the symmetry f(x)=f(1-x) reminds one of the Zeta function? My other comment was that \frac{d}{dx}e^{hx}F(x) made me think it looked like an integrating factor for some differential equation, but I think that was the most off-track of my comments. The key seems, to me, to be to determine where this integral, \int_0^1 h^{2n+1}e^{hx}f(x)\, dx comes from. Suggestions? I guess I should dig up the references…

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5 Responses to “Hardy and Wright, Chapter 4”

  1. sumidiot Says:

    I suppose the paper “Transcendence of e”, available here [pdf] (found via Google search: hermite proof e irrational) gives some justification for the sorts of things seen in HW’s proof about e being irrational.

    I’m going with: it was something terribly clever somebody wrote down. Though apparently not Euler, even if he did have a say [pdf].

    Another proof about e’s irrationality can be found here.

  2. Anonymous Says:

    The origin of these integrals is tied up with Pade approximants. See for example Section 3 in http://front.math.ucdavis.edu/0601.5660.

  3. sumidiot Says:

    Awesome! Thanks for the link. I’ll definitely check out that paper.

  4. Hardy and Wright, Chapter 5 « Sumidiot’s Blog Says:

    […] Sumidiot’s Blog The math fork of sumidiot.blogspot.com? « Hardy and Wright, Chapter 4 […]

  5. Hardy and Wright, Chapter 11 (part 2) « ∑idiot’s Blog Says:

    […] out a paper about the continued fraction expansion of (which maybe was pointed out to me in this comment), which talked about Pade approximations. Some of the things looked somewhat similar to what was […]

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